Field of the Invention
[0001] This invention relates to electronic warfare and in particular to a method for recognizing
observed electromagnetic signals emitted from modern radar systems, which are not
stable and do not correspond to emitter modes.
Background of the Invention
[0002] Electronic warfare (EW) is based upon the recognition of observed electromagnetic
signals, in particular, radar signals. The EW is an essential component of modern
warfare by providing information about, for example, the movement of enemy planes
or the launch of a rocket. Such Electronic Support (ES) functions allow, for example,
surveillance of enemy forces and warning of an imminent attack. Another aspect of
the EW is the Electronic Attack (EA) function such as jamming an enemy radar system
in order to substantially reduce the attack capability of an enemy force. A third
aspect of the EW is the Electronic Intelligence (ELINT) function that is concerned
with the interception and analysis of unknown radar signals for the population of
databases in order to support the ES and EA functions. The ELINT function is important
for ES and EA tactical systems, since these systems encounter in the field radar signals
emitted from unknown radar systems, or yet unknown signals emitted from known radar
systems.
[0003] In current EW systems, radar signals are recognized using histograms of observed
pulses in a parametric space - for example, frequency, pulse width, angle of arrival
- and temporal periodicities in the pulse train.
[0004] Andersen et al. describe in US Patent 5,583,505 issued Dec. 10, 1996 a radar pulse
detection and classification system that receives times-of-arrival of pulses from
simultaneous emitters, deinterleaves them into bands of pulse repetition intervals,
and determines the pulse periodicities using autocorrelation.
[0005] Caschera describes in US Patent 5,063,385 issued Nov. 5, 1991 a memory system for
histogramming the pulse descriptor word output of a radar warning receiver for quickly
determining the number and types of emitters of the observed radar signals.
[0006] In US Patent 4,918,455 issued Apr. 17, 1990, Maier teaches deinterleaving of sequential
signal pulses from unknown sources by clustering similar pulses into groups, and the
use of those groups to form hypothetical pulse train models.
[0007] Dunne et al. describe in US Statutory Invention Registration H513 published Aug.
2, 1998 a tracking apparatus using multi-processor modules for predicting in real
time the parametric behavior of radar signals to be jammed.
[0008] A major drawback of histogramming on parameters available on individual pulses is
that the temporal relationship of the pulses is lost in the histogram. Furthermore,
the periodic temporal analysis is limited by the assumption that the radar system
is a cyclo-stationary source of pulses. This holds true only for simple radar systems
over short periods of time.
[0009] Therefore, all these prior art EW systems - using histogramming and periodic temporal
analysis - are based upon the assumption that the observed electromagnetic signals
are stable and correspond to emitter modes. Emitter modes date back to the early days
of radar when an operator changed the signal by manually switching to another electrical
circuit. Therefore, the prior art EW systems are ill-suited for recognizing modern
"dynamic" radar systems. For example, in response to various events modern radar systems
change their emitted signal, which is automatically adjusted using a processor to
maximize radar performance. The signals are no longer stable and do no longer correspond
to emitter modes. Events causing a change of the emitted signal are, for example,
selection of a different range display by an operator, detection of a target by the
radar system and subsequently changing from a search to a tracking signal, switching
between a number of periodic signal patterns to reduce blind ranges and speeds, and,
launching of a missile triggering the transmission of a guidance signal from the radar
system.
[0010] It is, therefore, an object of the invention to overcome the drawbacks of the prior
art by providing a method capable of recognizing observed electromagnetic signals,
which are not stable and do not correspond to emitter modes.
[0011] It is further an object of the invention to provide flexibility in the modeling of
the radar system based upon the sensed signals and to preserve a maximum of information
provided by the sensed signals.
Summary of the Invention
[0012] The new method according to the invention provides the capability for recognizing
modern radar systems. Describing the radar system as a finite state automaton and
transforming it into a hidden Markov model provides flexibility and preserves a maximum
of information provided by the observed signals. The new method is compatible with
conventional receiver front-ends and allows integration into a wide range of legacy
ES, EA and ELINT systems. The only hardware requirement is a fast processor with sufficient
memory.
[0013] In accordance with the present invention there is provided a method for identifying
a source of electromagnetic signals comprising the steps of:
receiving an electromagnetic signal emitted from the source;
providing a finite state automaton for modeling the source, the finite state automaton
comprising a finite set of states and a set of transitions from state to state that
occur in dependence upon an input signal, the finite state automaton for producing
a sequence of output symbols from an output alphabet in dependence upon the state
transitions, such that the sequence of output symbols corresponds to the received
electromagnetic signal emitted from the source;
hidden Markov modeling of the finite state automaton and determining parameters of
the hidden Markov model such that a sequence of observation symbols produced from
an observation alphabet by the hidden Markov model is equal to the sequence of output
symbols; and,
identifying the source in dependence upon the determined parameters of the hidden
Markov model.
[0014] In accordance with an aspect of the present invention there is provided a method
for classifying source models in dependence upon an electromagnetic signal emitted
from a source of electromagnetic signals comprising the steps of:
receiving the electromagnetic signal emitted from the source;
providing a plurality of L source models λ such that the L source models λ have L different observation alphabets comprising symbols being integer multiples of L different time periods, the source models λ being hidden Markov modeled finite state
automatons;
transforming the received electromagnetic signal into L sequences of observation symbols O using the L different observation alphabets; and,
determining for each combination of a source model λ of the L source models with a sequence of observation symbols O(l) of the L sequences of observation symbols an observation probability P└λ(l)|O(l)┘, 1 ≤ l ≤ L of the source model λ for producing the sequence of observation symbols O(l).
[0015] In accordance with the aspect of the present invention there is further provided
a method for decoding an electromagnetic signal emitted from a source of electromagnetic
signals comprising the steps of:
receiving the electromagnetic signal emitted from the source;
determining a sequence of observation symbols O in dependence upon the received electromagnetic signal;
providing a source model λ, the source model λ being a hidden Markov modeled finite
state automaton;
determining a plurality of sequences of state transitions Q of the hidden Markov modeled finite state automaton; and,
determining for each of the sequences of state transitions Q a probability of occurrence P[Q|O,λ] with respect to the sequence of observation symbols O.
[0016] In accordance with the aspect of the present invention there is yet further provided
a method for predicting a second portion of an electromagnetic signal based upon a
first portion of the electromagnetic signal emitted from a source of electromagnetic
signals comprising the steps of:
receiving the first portion of the electromagnetic signal;
determining a partial sequence of observation symbols Ot in dependence upon the observed first portion of the signal;
providing a source model, the source model being a hidden Markov modeled finite state
automaton;
determining for each observation symbol of an observation alphabet of the source model
a probability for being the next symbol at time t + 1 in the sequence of observation
symbols O based on a state transition probability distribution at time t of the source model and an observation symbol probability distribution of the source
model at time t; and,
determining a most probable observation symbol.
[0017] In accordance with the aspect of the present invention there is yet further provided
a method for training a source model of a source emitting electromagnetic signals
comprising the steps of:
receiving an electromagnetic signal emitted from the source;
determining a sequence of observation symbols in dependence upon the received signal
emitted from the source;
providing the source model, the source model being a hidden Markov modeled finite
state automaton; and,
estimating a new source model based upon the probability ξt (i,j) of the source model being in state i at time t and in state j at time t + 1 for the sequence of observation symbols and the probability γt(i) of the source model being in state i at time t for the sequence of observation symbols.
Brief Description of the Figures
[0018] Exemplary embodiments of the invention will now be described in conjunction with
the following drawings, in which:
Figure 1 is a simplified transition diagram of a finite state automaton;
Figure 2 is a simplified diagram illustrating a hidden Markov model for generating
observation sequences produced by the finite state automaton of Fig. 1;
Figure 3 is a simplified diagram illustrating another hidden Markov model for generating
observation sequences produced by the finite state automaton of Fig. 1;
Figure 4 is a simplified diagram illustrating the hidden Markov model of Fig. 2 modified
to accept observation symbols of uniform duration;
Figure 5 is a simplified Trellis diagram derived from the hidden Markov model of Fig.
4;
Figure 6 is a simplified flow diagram of a method according to the invention for modeling
a radar system;
Figure 7 is a simplified flow diagram of a method according to the invention for classifying
radar models;
Figure 8 is a simplified flow diagram of a method according to the invention for decoding
an observed signal emitted from a radar system;
Figure 9 is a simplified flow diagram of a method according to the invention for predicting
a next observation symbol given a partial observation sequence; and,
Figure 10 is simplified flow diagram of a method for training a radar model according
to the invention.
Detailed Description of the Invention
[0019] A modern radar system is capable to change its pulse repetition interval (PRI) from
one stable value to another. Given sufficient observations it is possible to determine
the sequences of PRI changes that can occur and cannot occur, and the time intervals
separating the different PRIs. This information is very useful for recognizing a radar
system among others that use the same stable PRI values but in a different order or
with different intermediate intervals. For example, a radar system emits a stable
PRI followed by a sequence of pulses spaced by varying time intervals, the sequence
of intervals being periodic, exhibiting partial freedom, or being entirely pseudo-random.
[0020] In addition to changing the pulse-to-pulse interval many modern radar systems emit
different pulses. Recognizing the choice of pulse combined with the choice of interval
by the radar system provides further information for recognizing a radar system. For
example, two radar systems have a same set of pulses and intervals, but are still
distinguishable if the order they emit and space their pulses differs.
[0021] In order to provide the capability for recognizing modern radar systems a new method
according to the invention is disclosed. The new method describes each radar system
as a finite state automaton. This provides flexibility for modeling the radar system,
and preserves a maximum of information provided by the observed signals.
[0022] In the following, the method according to the invention will be explained in connection
with the recognizing of modern radar systems in EW applications, but is not limited
thereto. A person of skill in the art will find numerous applications of this method
for recognizing electromagnetic signals emitted from sources other than radar systems.
[0023] A finite state automaton is defined as follows: "A finite state automaton consists
of a finite set of states and a set of transitions from state to state that occur
on input symbols chosen from an alphabet. For each input symbol, there is exactly
one transition out of each state (possibly back to the state itself)." The definition
and a detailed description of finite state automata are found, for example, in Hopcroft,
J.E. and Ullman, J.D. "Introduction to Automata Theory, Languages and Computation"
Massachusetts, Addison-Wesley, (1979).
[0024] The input symbols and state transitions of the radar system are likely unobservable
by the EW system. What is observed by the EW system is the output of the finite state
automaton. The output is chosen from an alphabet containing, for example, pulses and
time intervals. The output is either associated with the state - called a Moore machine
- or with the transition - called a Mealy machine. The Moore machine and the Mealy
machine are equivalent, i.e. for any Moore machine there exists an equivalent Mealy
machine as shown, for example, in Hopcroft, J.E. and Ullman, J.D.
[0025] Referring to Fig. 1 the state diagram of a Moore machine is shown. A state diagram
is a directed graph with the vertices of the graph corresponding to the states of
the finite state automaton. If there is a transition from state
i to state
j on input
x, then there is an arc labeled from state
i to state
j in the state diagram. The finite state automaton described by the state diagram in
Fig. 1 has three states indicated by circles 1, 2, and 3, accepts a binary input from
an input alphabet {0, 1}, and produces a ternary output from an output alphabet {a,
b, c}. The outputs are shown above the states. If the automaton is in state 1 and
reads an input it necessarily enters state 2 because the corresponding arc is labeled
with both input symbols 0 and 1. Two arcs are leaving state 2, one labeled 0 going
to state 1, and one labeled 1 going to state 3. The next state, therefore, depends
on the input symbol read by the automaton. It is state 1 if the automaton reads an
input symbol 0 or it is state 3 if the automaton reads an input symbol 1. If the automaton
is in state 3 and reads an input symbol it goes to state 1. For example, starting
in state 1 and reading an input sequence of consecutive 0s, the automaton produces
the output sequence
ababab... . Starting in the same state and reading an input sequence of consecutive 1s,
it produces
abcabcabc... .
[0026] The finite state automaton representation of a radar system allows analysis of its
dynamic behavior for three reasons.
[0027] First, the dynamic behavior of the radar system is readily apparent upon examination
of the state transitions. Every time the finite state automaton is in a state from
which transition is possible to two or more states, the next state is selected based
on the input symbol. Therefore, the finite state automaton representation of the radar
system takes into account that the radar system is driven by events resulting in input
symbols that are randomly chosen, controlled by a tracking process, controlled by
an operator, etc. If the finite state automaton contains no state from which transition
is possible to two or more states the corresponding radar system is not dynamic, i.e.
it produces a periodic signal.
[0028] Second, the finite state automaton representation provides flexibility for modeling
a radar system by allowing the output alphabet to contain pulses, intervals, pulses
combined with intervals, bursts of consecutive pulses, portions of continuous wave
(CW) signals, etc. Furthermore, the pulses, intervals, etc. represented by the output
symbols need not to have a same duration.
[0029] Third, representing a radar system as a finite state automaton allows analysis using
hidden Markov models (HMM).
[0030] HMMs have been used very successfully for speech recognition. A detailed description
of the HMMs is given, for example, in Rabiner, Lawrence and Juang, Biing-Hwang "Fundamentals
of Speech Recognition", New Jersey: Prentice Hall (1993), and Jelinek, Frederick "Statistical
Methods for Speech Recognition (Language, Speech, and Communication)", Cambridge,
Mass.: MIT Press. (1999).
[0031] Jelinek and Frederick describe in "Continuous Speech Recognition by Statistical Methods",
Proc. IEEE, 64(4), 532-556, (1976) statistical methods of automatic recognition of
continuous speech concerning the modeling of a speaker and of an acoustic processor,
extraction of the models' statistical parameters, hypothesis search procedures, and
likelihood computations of linguistic decoding.
[0032] Levinson et al. disclose in US Patent RE 33,597 reissued in May 28, 1991, in a speech
recognizer storing of a set of signals each representative of a prescribed acoustic
feature and storing of a template for each spoken reference word which comprises signals
representative of a first state, a last state and a preselected number of intermediate
states of a constrained HMM.
[0033] Doddington et al. teach in US Patent 4,977,598 issued in Dec. 11, 1990 an efficient
pruning method which reduces computer processing unit loading during speech recognition
by means of a HMM.
[0034] Bahl et al. describe in US Patent 4,827,521 issued in May 2, 1989 training of Markov
models in a speech recognition environment, wherein word decoding accuracy is maximized
by maximizing the difference between the probability of the correct script of uttered
words given the label outputs and the probability of any other script.
[0035] An HMM is characterized by the following definitions, which will be used throughout
the disclosure of the invention:
1. N being the number of states in the model. The states are hidden, i.e. they are not
observable. The individual states are labeled as {1, 2, ..., N}, and the state of the model at time t is qt.
2. M being the number of distinct observation symbols per state, i.e. the discrete alphabet
size. The observation symbols correspond to the physical output of the system being
modeled. The individual symbols are denoted as V = {ν1, ν2, ..., νM}.
3. A = {aij} being a state transition probability distribution where

In the special case where each state reaches every other state of the model in a
single step, we have aij > 0 for all (i, j) pairs. In other cases we have aij = 0 for one or more (i, j) pairs.
4. B = {bj(k)} being an observation symbol probability distribution, in which

The observation symbol probability distribution defines the symbol distribution in
state j, j = 1, 2,..., N, and ot denotes the observation symbol at time t.
5. π = {πi} being an initial state distribution where πi = P[q1 = i],1 ≤ i ≤ N .
[0036] Given appropriate values of
N, M, A, B, and π the HMM generates an observation sequence
O = (
o1o2...
oT) with each observation
ot being one of the symbols from
V, and with
T being the number of observations in the sequence. For brevity, the compact notation
λ = (
A, B, π) will be used in the following to denote a complete parameter set of the HMM.
[0037] Based upon the above described characterization it is possible to model the states
and outputs of a finite state automaton as states and observation symbols in the HMM.
The main difference is that transitions are driven by input symbols in the finite
state automata while they occur statistically in HMMs. Therefore, every transition
out of every state - possibly back to the state itself - in the finite state automaton
is modeled as a non-zero transition probability in the probability distribution
A of the HMM. The lack of a transition in the finite state automaton is modeled as
a zero transition probability in the probability distribution
A. The difference between the finite state automaton and the HMM accommodates the fact
that the input symbols and state transitions are unobservable in EW applications.
[0038] Numerous HMMs are able to produce observation sequences explaining the output sequences
of a given finite state automaton. In the following three possible HMMs explaining
the output sequences of the Moore machine shown in Fig. 1 will be described.
[0039] Referring to Fig. 2 a possible HMM which is able to produce observation sequences
explaining the output sequences of the finite state automaton of Fig. 1 is shown.
The HMM of Fig. 2 has
N = 3 states. Each state corresponds to one of
M = 3 observation symbols, ν
1 =
a, ν
2=
b, and ν
3 =
c, respectively. The observation alphabet of this model is identical to the output
alphabet of the finite state automaton. A possible state transition probability
A is given by

where transition from state 1 is always to state 2, transition from state 3is always
to state 1, and transitions from state 2 to states 1 and 3 have been chosen to be
equiprobable.
[0040] Referring to Fig. 3 another possible HMM which is able to produce observation sequences
explaining the output sequences of the finite state automaton of Fig. 1 is shown.
The HMM of Fig. 3 has
N = 2 states. Each state corresponds to one of
M = 2 observation symbols, ν
1 = ab, and ν
2 =
abc, respectively. In this model the observation alphabet is different from the output
alphabet of the finite state automaton. A possible state transition probability distribution
A is given by

where all transition have been chosen to be equiprobable.
[0041] Before introducing the third HMM modeling for discrete time and uniform symbol duration
will be explained in the following.
[0042] In practice, most radar systems contain a clock, which is used to produce pulse to
pulse intervals as integer multiples of the clock period. Generally, it is possible
in EW systems to estimate the radar clock period, or a multiple thereof. In the following
this estimated period is denoted τ. For simplicity τ is assumed to be exact in following
description, but as is evident to a person of skill in the art the invention is not
limited thereto. Having estimated the radar period it is possible to transform the
observation signal into a sequence of observation symbols -
O = (
o1o2...
oT), wherein the duration of each symbol
ot is equal to an integer multiple of the clock period τ. This implies proper synchronization
and an observation alphabet
V where the symbol durations are all multiples of τ.
[0043] For modeling radar systems that do not use a clock, either τ is chosen to be very
small, or a continuous-time HMM is used.
[0044] To illustrate the discrete-time modeling the discrete-time output alphabet, shown
in Table 1, is chosen for the finite state automaton of Fig. 1.
Table 1
Symbol |
Duration |
Description |
a |
1τ |
a pulse |
b |
2τ |
Symbol a followed by an empty interval of one clock period |
c |
3τ |
Symbol a followed by an empty interval of two clock periods |
[0045] Denoting by 1 an interval containing a pulse, and by 0 an empty interval the output
alphabet is rewritten as
a = 1,
b = 10, and
c = 100.
[0046] Referring now to Fig. 4 the states and possible transitions of a third HMM of the
finite state automaton of Fig. 1 is shown, which accounts for output sequences produced
by a discrete-time finite state automaton. This model is essentially the HMM of Fig.
2 modified to accept observation symbols of uniform duration
τ. Defining
M = 2 observation symbols,
ν1 = 1 and
ν2 = 0, and then dividing each original state into one or more new states based on the
duration of the associated original symbol
N = 6 new states are obtained with the state transition probability distribution

[0047] Given the states and transitions of a uniform discrete-time HMM, it is possible to
construct a graph called the "Trellis diagram" as follows. First,
T copies of the
N states are replicated. The
N states of the
tth copy correspond to the HMM at time
t. A line with an arrow connects a state
Ai of the
tth copy with a state
Aj of the (
t + 1)th copy if the corresponding transition probability
aij is positive. For example, Fig. 5 shows the Trellis diagram for the HMM of Fig. 4.
The Trellis diagram leads to fast solution methods for many problems of interest.
[0048] In the models shown above a single observation symbol has been associated to each
state of the HMM. In the following we now consider the possibility of errors to occur
during the observation a signal emitted from the radar system. It is possible to consider
such errors in the HMM by means of the observation symbol probability distribution
B, which will be explained in the following using the third model, but is not limited
thereto.
[0049] For example, a pulse is observed when the radar output was an empty interval. Using
radar nomenclature we call this event a false alarm and denote the false alarm probability
as
PF. Conversely, it is possible that nothing is observed when the radar output was a pulse.
This event is called a miss and the miss probability is denoted
PM. Formally, we have


[0050] The corresponding observation symbol distribution
B = {
bj(
k)},
bj(
k) =
P[
ot =
νk|
qt =
j],
j = 1, 2, ..., 6, for the uniform discrete-time HMM of Fig. 4 is shown in Table 2.
Table 2
j |
k |
νk |
Bj(k) |
1 |
1 |
1 |
1 - PM |
1 |
2 |
0 |
PM |
2 |
1 |
1 |
1 - PM |
2 |
2 |
0 |
PM |
3 |
1 |
1 |
PF |
3 |
2 |
0 |
1 - PF |
4 |
1 |
1 |
1 - PM |
4 |
2 |
0 |
PM |
5 |
1 |
1 |
PF |
5 |
2 |
0 |
1 - PF |
6 |
1 |
1 |
PF |
6 |
2 |
0 |
1 - PF |
[0051] For brevity an HMM corresponding to a radar system is called the
radar model. We assume that the EW system contains
L competing radar models. The individual radar models are denoted as λ
(1),λ
(2),...,λ
(L). The models have different time periods, which are denoted as τ
(1),τ
(2),...,τ
(L). Hence, for a given observation signal, there are several observation sequences,
which are denoted as
O(1),
O(2),...,
O(L), respective to the different time periods.
[0052] Referring to Fig. 6, a method for modeling a radar system according to the invention
is shown. In a first step, the radar system is described as a finite state automaton
comprising a finite set of states and a set of transitions from state to state that
occur in dependence upon an input signal. The finite state automaton produces a sequence
of output symbols from an output alphabet in dependence upon the state transitions
such that the sequence of output symbols corresponds to an observed electromagnetic
signal emitted from the radar system. In a second step the finite state automaton
is transformed into a HMM such that a sequence of observation symbols produced from
an observation alphabet by the HMM is equal to the sequence of output symbols. Optionally,
given the states and transitions of a uniform discrete-time HMM, a Trellis diagram
is constructed.
[0053] For simplicity, discussion of the various concepts of the invention has been limited
to the observation of discrete symbols chosen from an alphabet. However, it is important
to note that it is possible to generalize all the disclosed concepts to continuously
distributed multivariate observations. This is achieved by replacing the observation
symbol probability distribution with a joint probability density function for which
a HMM training procedure has been formulated. Compliant joint density functions include
elliptically symmetric density functions like multivariate Gaussians and multivariate
mixtures like Gaussian sums. In EW, generalization to multivariate observations allows
use of parameters like pulse width and frequency.
[0054] In modern electronic warfare four basic problems are encountered.
[0055] The first problem is the classification problem: given an observed signal and
L competing radar models λ
(1), λ
(2),...,λ
(L), how is the model chosen which best matches the observed signal?
[0056] The second problem is the decoding problem: given the observation sequence
O and the radar model λ, how is a state sequence

chosen which explains best the observations?
[0057] The third problem is the prediction problem: given the partial observation sequence
(
o1o2...
ot) and the radar model λ, how is the next observation symbol

= ν
k predicted and the associated probability
P[
ot+1 = νk|
o1o2...ot, λ] computed?
[0058] The fourth problem is the training problem: given only the observation sequence
O, how is the radar model λ adjusted - or trained - so that it best describes the generation
of the observed signals produced by the radar system?
[0059] The classification problem is critical for ES and EA. Current approaches use histograms
of the pulses in a parametric space and temporal periodicities in the pulse train,
extract parameters and match them to competing entries in a library. When the extracted
parameters fall into an overlap area of two or more entries of the library, the emitter
is not uniquely identified, resulting in an ambiguity. EW systems are designed to
communicate multiple candidate identifications to the operator. For example, a friendly
emitter leads to multiple candidate identifications that include a threat. This triggers
an unnecessary alert, jeopardizing safety and increasing the risk of fracticide. Since
the method for modeling a radar system according to the invention preserves more information
about the radar system than current approaches, it has the potential to substantially
enhance the classification of radar systems.
[0060] Since the
L competing radar models have different time periods the observation signal is transformed
into
L sequences of observation symbols,
O(1),
O(2), ...,
O(L), where the duration of each symbol
ot(l) of the
lth sequence is equal to an integer multiple of the
lth time period τ
(l), 1≤
l≤
L.
[0061] Formally, the classification problem now becomes: given
O(1),
O(2), ...,
O(L) and λ
(1),λ
(2),...,λ
(L), how is the most probable of the
L radar models chosen?
[0062] From Bayesian decision theory, one of the most useful decisions is to maximize the
a posteriori probability. The maximum a posteriori (MAP) decision is to choose

[0063] Using Bayes rule, we have

[0064] Since
P[
O] is not a function of λ, the MAP decision is also defined as

[0065] However, the MAP decision requires prior knowledge of the probabilities
P└λ
(l)┘, 1≤
l≤
L, which is not always available.
[0066] When the probabilities
P[λ
(l)], 1 ≤
I ≤
L, are not known a priori, it is assumed that the competing radar models are equiprobable.
Then the maximum likelihood (ML) decision is used which is defined as follows

[0067] Whether the MAP or ML decision is used for each of the
L competing radar models the probability of the sequence for a given model is computed.
This problem is called the evaluation problem: given an observation sequence
O = (
o1o2...
oT) and a radar model λ, how is
P[
O|λ] - the probability of the sequence for a given radar model - computed?
[0068] One possibility is a direct approach to solving the evaluation problem. However,
since computing the probability of
O given a state sequence
Q requires the order of
T calculations, and since there are
NT possible state sequences, a direct computation of
P[
O|λ] requires in the order of
TNT calculations. Because of the large number of calculations and its non-polynomial
complexity, the direct approach is practical only for small HMMs.
[0069] A method for solving the evaluation problem for large HMMs is the forward-backward
procedure which is described in detail, for example, in Baum, L. E. and Egon, J. A.
"An Inequality with Applications to Statistical Estimation for Probabilistic Functions
of a Markov Process and to a Model for Ecology", Bull. Amer. Meteorol. Soc., 73, 360-363,
(1967), and Baum, L. E. and Sell, G. R. "Growth Functions for Transformations on Manifolds",
Pac. J. Math., 27(2), 211-227, (1968). The solution of the evaluation problem using
the forward-backward procedure requires in the order of
N2T calculations.
[0070] Consider the forward variable

i.e. the probability of the model being in state
i at time
t and observing the partial sequence (
o1o2...
ot), for a given model λ. The forward variable at time
t = 1 is the joint probability of state
i and observation
o1,

[0071] At time
t = 2 the forward variable is

[0072] Proceeding inductively, we obtain for 1 ≤
t ≤
T -1,

[0073] Finally, the sum of the terminal forward variables provides the solution to the evaluation
problem, i.e.

[0074] The forward-backward procedure owes its efficiency to the Trellis representation
of uniform discrete-time HMMs.
[0075] Once the evaluation problem has been solved it is possible to compute for each of
the
L models in the classification problem the probability of the sequence for each given
model and to decide between the models using the MAP or the ML criterion.
[0076] A special situation occurs when the ES or EA system encounters in the field either
an unknown radar system, or an unknown signal produced by a known radar system. When
such a situation occurs it is not desired that the ES or EA system chooses a model
of its library for the unknown signal, but provides an indication that an unknown
signal has been observed. This results in a decision with a reject option. Formally,
the decision problem has then
(L + 1) values: 1, 2,...,
L, and "reject".
[0077] Referring to Fig. 7 a simplified flow diagram of a method for classifying radar models
according to the invention is shown.
[0078] The decoding problem is: given the observation sequence
O and the radar model λ, how is a state sequence

chosen which explains best the observations?
[0079] Particularly useful for ES and EA applications is not so much the complete state
sequence

, as the recognition of specific state transitions providing situational information.
For example, when a missile is launched, the associated radar system often changes
its signal to guide the missile. In order for the radar signal to change, some state
transition has to occur in the finite state automaton of the radar. By observing the
state transition that corresponds to a missile launch, the ES or EA system is able
to recognize the event and to warn that a missile is airborne. In another example,
when a radar senses a new target it often changes its signal to estimate the position
and velocity of the target. Again in order for the radar signal to change, some state
transition has to occur in the finite state automaton of the radar. By observing this
state transition the ES or EA system is able to warn that a target is being acquired.
[0080] Assuming that the radar system has already been classified, the motivation behind
the decoding problem is to recognize changes in the radar signal and, ultimately,
to infer what events are happening. If the HMM contains the state transitions that
correspond to critical events it substantially enhances situational awareness.
[0081] The solution to the decoding problem depends on the choice of an optimality criterion
to determine which sequence is the best. The optimality criterion is to maximize
P[
Q|
O,λ]. This criterion accounts for the probability of occurrence of sequences of states.
Therefore, it yields a sequence for which all state transitions have positive probability,
i.e. a valid sequence.
[0082] For solving the decoding problem the dynamic programming method - also known as the
Viterbi method - is used. Detailed information about this method is provided, for
example, in Bellman, Richard "Dynamic Programming", New Jersey, Princeton University
Press, (1957), and in Bertsekas, Dimitri "Dynamic Programming and Optimal Control"
2 ed. Belmont, MA: Athena Scientific, (2000). Optionally, less optimal methods such
as the stack method are also applicable for solving the decoding problem. In particular,
if exhaustive exploration of the Trellis diagram is prohibitive.
[0083] Fig. 8 illustrates a simplified flow diagram of a method for decoding an observed
signal emitted from a radar system according to the invention.
[0084] The prediction problem is: given the partial observation sequence (
o1o2...
ot) and the radar model λ, how is the next observation symbol

= ν
k predicted and the associated probability
P[
ot+1 = ν
k|
o1o2...
ot,λ] computed?
[0085] This problem is of particular interest for EA systems using deceptive electronic
countermeasures techniques. For example, the range gate pull-off technique involves
the transmission of false echo pulses just before the arrival of the actual radar
pulses to create a false target. To simulate an inbound target the jammer anticipates
the reception of the radar pulse and transmits a false echo pulse before the radar
pulse. For this technique to work it is crucial that the time of arrival of the radar
pulse is predicted reliably and accurately.
[0086] This problem is closely related to the evaluation problem and similarly solved. Using
the forward variable the probability that the next symbol is ν
k is

[0087] Therefore, the most probable symbol ν
k to occur at time
t + 1 is

[0088] Similarly the HMM is used to predict the most probable symbol to occur at time
t + 2,
t + 3, etc. Furthermore, it is possible to use the HMM in a similar fashion to estimate
the time a given symbol is most likely to be observed next.
[0089] A simplified flow diagram of a method according to the invention for predicting a
next observation symbol given a partial observation sequence is shown in Fig. 9.
[0090] The training problem is: given only the observation sequence
O, how is the radar model λ adjusted - or trained - so that it best describes the generation
of the observed signals produced by the radar system?
[0091] Prior to solving the classification, decoding and prediction problems radar models
are needed. Training an HMM is viewed as a reverse engineering problem: knowing what
the radar produces, what is the finite state automaton in the radar model? The training
problem is of particular interest for ELINT, a branch of the EW concerned with the
interception and analysis of radar signals for the population of databases.
[0092] Ideally, the training problem is defined as finding the maximum a posteriori radar
model

[0093] However, the solution to this equation is most likely not unique. In practice we
have only partial knowledge of the probabilities
P[λ] for all possible λ s and the search space comprising all possible λ s is very
large.
[0094] Not knowing the number
N of states different values of
N are used and then for each given
N the corresponding MAP model is searched, and finally the best of the models found
is chosen.
[0095] To illustrate the difficulty of the problem we return to the Moore machine of Fig.
1. Considering that it has produced an output
abcabcabab, or rewritten using the discrete-time alphabet 110100110100110110. Assuming
M = 2, ν
1 = 1, ν
2 = 0, and that no observation error has occurred, then the observation sequence
O is the same as the Moore machine's output.
[0096] In the following, two very simple HMMs will be examined and compared. For the smallest
possible number of states,
N = 1, we arrive at a model defined by
A = [1],
b1(1) =

,
b1(2) =

, and π
1 = 1. The probability of this radar model λ
(1) producing the sequence
O is

[0097] For the number of states being equal to the number of observation symbols of the
observation sequence
O, N = 18, we arrive at the model defined by
A being equal to the
N ×
N diagonal matrix,
bt(1) = 1 and
bt(2) = 0 for
t such that
ot = 1, and
bt(1) = 0 and
bt(2) = 1 for
t such that
ot = 0, π
1 = 1, and π
i = 0 for 2 ≤
i ≤ 18. The probability of this radar model λ
(2) producing the sequence
O is

[0098] Which of the two models λ
(1) or λ
(2) should be chosen? The latter is more likely to have produced the sequence
O, but has a number of states equal to the number of observation symbols of the observation
sequence
O. However, is the latter model plausible? Answering such a question requires prior
knowledge, for example, from radar engineering. Consequently, the success of automatic
HMM training depends as much on the use of prior knowledge in the form of
P[λ], as on the maximization of the probability of the sequence being observed from
the model
P[
O|λ].
[0099] Given a lack of prior statistics the maximum likelihood model

is considered, which is considerably simpler than the MAP problem. Still, there is
no known way for analytically solving the ML problem.
[0100] Assuming knowledge of the number of states
N the ML problem is iteratively solved using the Baum-Welch method, which is, for example
discussed in Baum, L.E. "An Inequality and Associated Maximization Technique in Statistical
Estimation for Probabilistic Functions of Markov Processes", Inequalities, 3, 1-8,
(1972). The solution of the ML problem leads to local maxima of the probability of
the sequence being observed from the model,
P[
O|λ]. In most problems of interest, the optimization surface is complex and has many
local maxima and, therefore, the solution found, λ
BW, is not necessarily the optimum.
[0101] Before describing the iterative solution, we first define the backward variable as

i.e. the probability of observing the partial sequence (
ot+1ot+2 ...
oT) for a given state
i at time
t and a given model λ. The backward variable at time
T is arbitrarily defined as

[0102] Proceeding inductively, we obtain for 1 ≤
t ≤
T - 1,

[0103] We define the probability of being in state
i and at time
t and state
j at time
t + 1, given the observation sequence and the model, as

[0104] Using the definitions of the forward and backward variables ξ
t (
i,
j) is written in the form


[0105] Next we define the variable

as the probability of being in state
i at time
t, given the observation sequence and the model. Using the definitions of the variables
α
t (
i), β
t (
i), and ξ
t(
i,j), γ
t(
i) is rewritten in the form

[0106] If we sum
γt (
i) over time, we get the expected number of transitions from state
i. Similarly, if we sum ξ
t(
i,
j) over time, we get the expected number of transitions from state
i to state
j.
[0107] Based on the above variables and formulas, the following equations are used to iteratively
estimate π,
A, and
B:



wherein
t* is such that
ot = ν
k.
[0108] Denoting the current model as λ = (
A, B, π) equations (25) - (27) are then used to estimate a new model

= (

,

,

). It has been proven that if

≠λ, then the probability of
O being observed from the model has increased, i.e.
P└
O|

┘>
P[
O|
λ]. In a following iteration step the current model is set equal to the new model and
equations (25) - (27) are then used again to reestimate the model. If

= λ a limit has been reached and the model

= λ
BW corresponds to a local maximum of
P[
O|λ].
[0109] Again the optimization surface is very complex and has many local maxima. Therefore,
the model λ
BW found by the iterative solution depends to a large extent on the initial condition,
i.e. the radar model λ defined as current in the first estimation.
[0110] Referring to Fig. 10 a simplified flow diagram of a method for training a radar model
according to the invention is shown.
[0111] The new method according to the invention provides the capability for recognizing
modern radar systems. Describing the radar system as a finite state automaton and
transforming it into a HMM provides flexibility and preserves a maximum of information
provided by the observed signals. The new method is compatible with conventional receiver
front-ends and allows integration into a wide range of legacy ES, EA and ELINT systems.
The only hardware requirement is a fast processor with sufficient memory.
[0112] Numerous other embodiments of the invention will be apparent to persons skilled in
the art without departing from the spirit and scope of the invention as defined in
the appended claims.